Relations & Functions Questions - JEE

If $x \in (0, \frac{\pi}{2})$, $\tan x \in (0, \infty)$, find the minimum value of the function $f(x) = 3\tan x + \cot x$.

If $f(x)$ satisfies the relation $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$ and $f(1) = 5$, then find $\displaystyle\sum_{n=1}^{m} f(n)$. Also prove that $f(x)$ is odd.

Consider the following two statements:P: If 7 is an odd number, then 7 is divisible by 2.Q: If 7 is a prime number, then 7 is an odd number.If $V_1$ is the truth value of the contrapositive of $P$ and $V_2$ is the truth value of contrapositive of $Q$, then the ordered pair $(V_1, V_2)$ equals

For the function with domain $D: [-2,-1] \cup [1,2]$ and $-1 \leq \log_2\left(\dfrac{x^2}{2}\right) \leq 1$, find the range.

Let $R$ be a relation defined on the set of all natural numbers as $R = \{(x, y) : x \in \mathbb{N}, 2x + y = 41\}$. Find the number of elements in the set domain of this relation.

If $n(A) = 3$, $n(B) = 6$ and $A \subseteq B$. Then the number of elements in $A \cup B$ is equal to

Let $f$ satisfy $f(10+x)=f(10-x)$ and $f(20+x)=-f(20-x)$ for all $x\in\mathbb{R}$. Which statement is correct?

Suppose A1, A2, ..., A30 are thirty sets each having 5 elements and B1, B2, ..., Bn are n sets each having 3 elements. Let $\bigcup_{i=1}^{30} A_i = \bigcup_{j=1}^{n} B_j = S$ and each element of S belongs to exactly 10 of the Ai's and exactly 9 of the Bj's, then find the value of n.

If the functions $f(x) = e^x/a$ and $g(x) = \ln(ax)$ are inverse of each other, then the value of $[a]$ (where $[\cdot]$ denotes the greatest integer function) is:

Given that $ \dfrac{1}{|x| - 3} \leq \dfrac{1}{2} $, the solution set is:

The number of functions $f: S \to S$, where $S = \{1, 2, 3\}$, such that $f[f(x)] = f(x)$ for all $x \in S$ is:

If $p$, $q$ and $r$ are simple propositions such that $(p \wedge q) \wedge (q \wedge r)$ is true, then

137. If $g(x)$ and $h(x)$ are invertible functions and $h(x)=3g(x)+7$, then $h^{-1}(x)$ is equal to:

$ S = \{1, 2, 3\} $, $ f : S \to S $ satisfies the property: $ \forall x \in S,\, f(f(x)) = f(x) $. How many different functions are there for $ f(x) $?[Note: Can you generalize the result for $ S = \{1, 2, 3, \ldots, n\} $?]

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to

It is given that $ f(x+y) = f(x)\,f(y) $ and $ f(1) = 2 $. If $ \displaystyle\sum_{k=1}^{n} f(a+k) = 16(2^n - 1) $, find the value of $ a $.

Find the domain of the function $f(x) = \dfrac{1}{\sqrt{|x|^2 - |x| - 6}}$.

Let $P(x) = kx^3 + 2k^2x^2 + k^3$. If $(x-2)$ is a factor of $P(x)$, find the sum of all real values of $k$.

Let A and B be two non-empty subsets of a set X such that A is not a subset of B. Then

Contrapositive of the statement 'If two numbers are not equal, then their squares are not equal' is:

If $f(x) = \left(\dfrac{3}{5}\right)^x + \left(\dfrac{4}{5}\right)^x - 1$, $x \in \mathbb{R}$, then the equation $f(x) = 0$ has

The graph of the function $f(x) = \frac{9x+7}{3x+12}$ is symmetric to the point:

Consider the following statements:$p$ : It rains today$q$ : I go to school$r$ : I shall meet any friends$s$ : I shall go for a movieThen which of the following propositions represents 'If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie.'?

Let $f(x)$ and $g(x)$ are two functions defined from $R^+ \to R$ such that\[f(x) = \begin{cases} 1 - \sqrt{x}, & \text{if } x \text{ is rational} \\ x^2, & \text{if } x \text{ is irrational} \end{cases}\] and \[g(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 1 - x, & \text{if } x \text{ is irrational} \end{cases}\]The composite function $f(g(x))$ is:

The domain of the function $f(x) = \dfrac{1}{\sqrt{|x| - x}}$ is

Let $f(x) = 2^{10} \cdot x + 1$ and $g(x) = 3^{10} \cdot x - 1$. If $(f \circ g)(x) = x$, then $x$ is equal to

$\sim(p \vee (\sim p \vee q))$ is equal to

If $f$ is a function with domain $[-3, 5]$ and $g(x) = |3x + 4|$, then the domain of $(f \circ g)(x)$ is:

Which one of the following Boolean expressions is a tautology?

The graph of the function $y = f(x)$ is symmetrical about the line $x = 2$, then

JM Q26.

Let R be the real line. Consider the following subsets of the plane $R \times R$:\(S = \{(x, y) : y = x + 1 \text{ and } 0 Which one of the following is true?

Given $ f(x) = \left|1 - \dfrac{1}{x}\right| $. Which of the following statements is correct about f(x)?

JM Q33.

Let $f(x)=x+3$ for $x\in\mathbb{Q}$, $4x$ for $x\in\mathbb{R}\setminus\mathbb{Q}$; and $g(x)=\sqrt{5}+x$ for $x\in\mathbb{R}\setminus\mathbb{Q}$, $-x$ for $x\in\mathbb{Q}$. Find the nature of $(f-g)(x)$.

Let $f(x)=2x-\{x/\pi\}$, $g(x)=\cos x$. Find the period of $(g\circ f)(x)$.

Consider the two sets A = \{m \in \mathbb{R} : \text{both the roots of } x^2 - (m+1)x + m + 4 = 0 \text{ are real}\} and B = [-3, 5). Which of the following is not true?

The relation R defined on the set of natural numbers as \{(a, b) : a \text{ differs from } b \text{ by } 3\} is given by

$f(x) = \sin \frac{x}{6} + \cos \frac{7x}{10}$. Period of $\sin \frac{x}{6} = 6\pi$. Period of $\cos \frac{7x}{10} = \frac{20\pi}{7}$. $\sin a\theta, \cos a\theta$ are periodic with period $\frac{2\pi}{a}$. $\text{LCM}(6\pi, \frac{20\pi}{7}) = 60\pi$. Therefore, the period of $f(x) = 60\pi$. Hence, $n = 6$.

For x \in \mathbb{R}, the function f(x) satisfies 2f(x) + f(1-x) = x^2. The value of f(4) is equal to

Let g: [\frac{π}{2}, π] → A defined by g(x) = \frac{\sin x + 4}{\sin x - 2} be an invertible function. Find the set A.

If $f(x) = \log_e\left(\dfrac{1-x}{1+x}\right)$, \(|x|

Let $g(x) = x^2 + ax + b$ $h(x) = cx - x^2$ where $1 + a + b = 0$ and $1 - c = 0$. If $g'(1) = h'(1)$, find the value of $a \cdot b \cdot k$ where $k$ is determined by the condition. (Find $b$)

Match number of integers in each set with (P)0 (Q)2 (R)3 (S)less than 3 (T)more than 3

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that $Y \subseteq X$, $Z \subseteq X$ and $Y \cap Z$ is empty, is

Let $f(x)=x^2+3x+2$. Which are correct?(A) $f(|x|)=2$ has 1 solution (B) 3 solutions (C) $|f(x)|=0.125$ has 4 solutions (D) $|f(|x|)|=0.125$ has 8 solutions

$S = \{1, 2, 3\}$, $f : S \to S$ satisfies the property: $\forall x \in S,\; f(f(x)) = f(x)$. How many different functions are there for $f(x)$?[Note: Can you generalize the result for $S = \{1, 2, 3, \ldots, n\}$?]

$(D): A = [-\frac{3}{2}, \frac{3}{2}], B = [2\sqrt{2}, 4\sqrt{2}]$

If $f(x)$ is a periodic function with period $a$ and $f(-1) = f(a-1)$, and $f(x) = (x-1)^2 - 6(x-1) + 8$, then the value of $a$ (other than 0) is:

Let $f,g:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=|x-1|$ and $g(x)=\begin{cases}e^x, & x\geq0\\ x+1, & x\leq0\end{cases}$. Then the function $f(g(x))$ is:

Relations & Functions Questions (810)